fukuda equivalence

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arXiv:quant-ph/0608010v1

1Aug2006

Simplification of additivity conjecture in quantum

information theory

Motohisa Fukuda

Statistical Laboratory,

Centre for Mathematical Sciences,

University of Cambridge

February 3, 2007

Abstract

We simplify some conjectures in quantum information theory; the additivityof minimal output entropy, the multiplicativity of maximal output p-norm andthe superadditivity of convex closure of output entropy. In this paper, by usingsome unital extension of quantum channels, we show that proving one of theseconjectures for all unital quantum channels would imply that it is also true for allquantum channels.

1 Introduction

It is natural to measure the noisiness of a (quantum) channel by the minimal ouputentropy (MOE); how close can the output be to a pure state in terms of the vonNeuman entropy. It is then important to ask if a tensor product of two channelscan ever be less noisy in the sense that some entangled input can have its outputcloser to a pure state than the product of the optimal inputs of the two channels.This leads to the additivity conjecture of MOE. Actually, the additivity of MOE

has been shown [3],[20],[18] to be equivalent globally to several other fundamen-tal conjectures in quantum information theory; the additivity of Holevo capacity,the additivity of entanglement of formation and the strong superadditivity of en-tanglement of formation. In this papar we write not only about the additivity of

Email: [email protected]

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MOE but also about two other conjectures; the multiplicativity of maximal output

p-norm, which measures how close can the output be to a pure state in terms ofp-norm, and the superadditivity of convex closure of output entropy. Note thatthe multiplicativity of maximal output p-norm implies the additivity of MOE [10],and the superadditivity of convex closure of output entropy the additivity of en-tanglement of formation [16],[20]. In this sense, these two conjectures are strongerthan the above four equivalent conjectures.

Curiously, most channels for which the additivity of MOE has been provenare unital: unital qubit channels [11], the depolarizing channel [8],[13],[2], theWerner-Holevo channel [17],[4],[1], the transpose depolarizing channel [7],[5], andsome asymmetric unital channels [6]. By contrast, non-unital channels have beenextremely resistant to proofs. Of course there are some proofs on non-unital chan-nels; entanglement-breaking channels [19],[12], a modification of the Werner-Holevochannel [22] and diagonal channels [14]. However in the paper [9] we had to extendthe result on the depolarizing channel, which is unital, to non-unital ones.

In this paper we simplify the three conjectures; the additivity of MOE, themultiplicativiy of maximal output p-norm and the superadditivity of convex closureof output entropy. We show that proving these conjectures for a product of anytwo unital channels is enough by using some unital extension of channels. This issignificant because having proven this we dont have to consider non-unital channelsas long as these conjectures are concerned.

Let us give some basic definitions. A (quantum) state is represented as apositive semidefinite operator of trace one in a Hilbert space H; this is calleda density operator. We denote the sets of all bounded operators and all density

operators in H by B(H) and D(H) respectively. A (quantum) channel from H1to H2 is a completely positive (CP) trace-preserving (TP) map (CPTP map) fromB(H1) to B(H2). A channel is called bistochastic if

(IH1) = IH2 .

Here IH1 = IH1/dimH1, called the normalised identity, where IH1 is the identityoperator in H1 (IH2 is similarly defined). When H1 = H2 bistochastic channelsare called unital channels.

The MOE of a channel is defined as

Smin() := infD(H)

S(()), (1.1)

where S is the von Neumann entropy: S() = tr[ log ]. The additivity conjec-ture of MOE [15] is that

Smin( ) = Smin() + Smin(), (1.2)

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for any channels and . Note that the bound Smin( ) Smin()+ Smin()

is straightforward.The maximal output p-norm of a CP map is defined as

p() := supD(H)

()p, (1.3)

where p is the Schatten p-norm: p = (tr||p)

1

p . The multiplicativity conjec-ture of maximal output p-norm is that

p( ) = p()p(), (1.4)

for any CP maps and , and any p [1, 2]. The multiplicativity was conjectured

to be true for p [1, ] before a counterexample was found [21]. Note that thebound p( ) p()p() is straightforward.The convex closure of output entropy of a channel and a state can be

written as

H() = min

i

piS(i) :i

pii = ,i

pi = 1, pi 0

. (1.5)

The superadditivity conjecture of convex closure of output entropy is that

H() H(H) + H(K), (1.6)

for any channels and , and any state D(H K). Here the input spaces of and are H and K respectively, and H = trK[] and K = trH[].We introduce the Weyl operators to be used later. Given a Hilbert space H of

dimension d let us choose an orthonormal basis {ek; k = 0, . . . , d1}. Consider theadditive cyclic group Zd and define an irreducible projective unitary representationof the group Z = Zd Zd in H as

z = (x, y) Wz = UxVy,

where x, y Zd, and U and V are the unitary operators such that

U|ek = |ek+1(modd), V|ek = exp2ikd |ek.

Then we have z

WzWz = d

2IH, D(H). (1.7)

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2 Result

We were inspired by the Shors paper [20] to find the following theorem.Theorem 1. Take a channel .

1) The additivity of MOE of b for any bistochastic channel b would implythat of for any channel .2) Fix p [1, ]. The multiplicativity of maximal output p-norm of b forany bistochastic channel b would imply that of for any CP map .3) The superadditivity of convex closure of output entropy of b for anybistochastic channel b would imply that of for any channel .

Proof. 1) Suppose we have a channel

: H1 H2.

Here the dimension of H2 is d. Then we construct a new channel :

: Cd2

H1 H2

z

Wz(EzEz )W

z .

Here Wz are the Weyl operators in H2 and Ez = (z| IH1), where {|z} formsthe standard basis for Cd

2

. Note that this channel is bistochastic by (1.7).First, we show

Smin( ) Smin( ), (2.1)

for any channel . Suppose is a channel from K1 to K2. Then

( )(|(0, 0)(0, 0)| )

= (1H2 )(( 1K1)(|(0, 0)(0, 0)| ))

= (1H2 )(( 1K1)())

= ( )(),

for any D(H1 K1).Next, we show the converse;

Smin( ) Smin( ), (2.2)

for any channel . Take D(Cd2

H1 K1). Let z = (Ez IK1)(Ez IK1),

cz = trz and z = z/cz. Note that can be written in the matrix form:

=

(0,0) . . . ...

. . ....

. . . (d1,d1)

.

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This is a d2 d2 block matrix, where each block is an element in B(H1 K1).

Then, by concavity of the von Neumann entropy,

S

( )()

= S

z

(Wz IK2)(( )(z))(Wz IK2)

z

czS(( )(z))

Smin( ).

Finally, since we assumed the additivity for a product of any unital channeland we get, by using (2.1) and (2.2),

Smin( ) = Smin(

) = Smin(

) + Smin() = Smin() + Smin().

2) When is a CP map the extension is such that (IH1) = cIH2 for somepositive constant c. The multiplicativity for /c, which is a bistochastic channel,would imply that for .

3) As in the proof 1) take any channel to have the following result:

H(|(0, 0)(0, 0)| ) = mini

piS(( )(|(0, 0)(0, 0)| i))

= mini

piS(( )(i))

= H() D(H1 K1).

To see the first equality note that

|(0, 0)(0, 0)| =i

pii i = |(0, 0)(0, 0)| i i.

By the assumption we have

H() = H(|(0, 0)(0, 0)| )

H(|(0, 0)(0, 0)| H1) + H(K1)

= H(H1) + H(K1)

QEDRemark. The first part of proof 1) shows that is a bistochastic extension

of . In the following corollary we form a unital extension of .Corollary 2. In each case of theorem 1, the assumption would be implied by

proving the conjecture on u for all unital channels u.

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Proof. Consider a unital channel:

: Cd2 H1 Ccd H2

ICcd ().

Here c is the dimension of H1. Then it is not difficult to see

S(( )()) = log cd + S(( )())

( )()p = (cd)1p

p ( )()p,

for any channel and any state D(Cd2

H1K1). The results follow obviously.QED

Corollary 3. The following statements are true.

1) The additivity of MOE of u u for any unital channels u and u wouldimply that of for any channels and .2) Fix p [1, ]. The multiplicativity of maximal output p-norm of u u forany unital channels u andu would imply that of for any CP maps and.3) The superadditivity of convex closure of output entropy of u u for anyunital channels u and u would imply that of for any channels and .

3 Conclusion

By using the results in this paper we can focus on unital channels to prove theadditivity of MOE, the multiplicativity of maximal output p-norm and the super-additivity of convex closure of output entropy, or to find a counterexample.

Acknowledgement

I would like to thank my supervisor Yuri Suhov for suggesting the problem, con-stant encouragement and numerous discussions. I also would like to thank Alexan-der Holevo for giving useful comments and especially pointing out that the theoremalso works for the superadditivity of convex closure of output entropy.

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